{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,7,30]],"date-time":"2025-07-30T16:43:41Z","timestamp":1753893821764,"version":"3.41.2"},"reference-count":0,"publisher":"The Electronic Journal of Combinatorics","issue":"1","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Electron. J. Combin."],"abstract":"<jats:p>Let $m$ be a positive integer and let $G$ be a graph. A set ${\\cal M}$ of matchings of $G$, all of which of size $m$, is called an $[m]$-covering of $G$ if $\\bigcup_{M\\in {{\\cal M}}}M=E(G)$. $G$ is called $[m]$-coverable if it has an $[m]$-covering. An $[m]$-covering ${\\cal M}$ such that $|{{\\cal M}}|$ is minimum is called an excessive $[m]$-factorization of $G$ and the number of matchings it contains is a graph parameter called excessive $[m]$-index and denoted by $\\chi'_{[m]}(G)$ (the value of $\\chi'_{[m]}(G)$ is conventionally set to $\\infty$ if $G$ is not $[m]$-coverable). It is obvious that $\\chi'_{[1]}(G)=|E(G)|$ for every graph $G$, and it is not difficult to see that $\\chi'_{[2]}(G)=\\max\\{\\chi'(G),\\lceil |E(G)|\/2 \\rceil\\}$ for every $[2]$-coverable graph $G$. However the task of determining $\\chi'_{[m]}(G)$ for arbitrary $m$ and $G$ seems to increase very rapidly in difficulty as $m$ increases, and a general formula for $m\\geq 3$ is unknown. In this paper we determine such a formula for $m=3,$ thereby determining the excessive $[3]$-index for all graphs.<\/jats:p>","DOI":"10.37236\/213","type":"journal-article","created":{"date-parts":[[2020,1,11]],"date-time":"2020-01-11T04:24:42Z","timestamp":1578716682000},"source":"Crossref","is-referenced-by-count":7,"title":["The Excessive [3]-Index of All Graphs"],"prefix":"10.37236","volume":"16","author":[{"given":"David","family":"Cariolaro","sequence":"first","affiliation":[]},{"given":"Hung-Lin","family":"Fu","sequence":"additional","affiliation":[]}],"member":"23455","published-online":{"date-parts":[[2009,10,5]]},"container-title":["The Electronic Journal of Combinatorics"],"original-title":[],"link":[{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v16i1r124\/pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v16i1r124\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2020,1,18]],"date-time":"2020-01-18T02:40:24Z","timestamp":1579315224000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/view\/v16i1r124"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2009,10,5]]},"references-count":0,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2009,1,7]]}},"URL":"https:\/\/doi.org\/10.37236\/213","relation":{},"ISSN":["1077-8926"],"issn-type":[{"type":"electronic","value":"1077-8926"}],"subject":[],"published":{"date-parts":[[2009,10,5]]},"article-number":"R124"}}